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1 Arithmetic mean:

(i) For ungrouped data (individual series) x =

(ii) For grouped data (continuous series)

(a) Direct method x = where xi, I = 1 .. n be n observations and fi be their corresponding

frequencies

(b) short cut method : x = A + fidi / fi) where A = assumed mean, d i = xi - A = deviation for each term

2 Properties of A.M.

(i) In a statistical data the sum of the deviation of items form A.M. is alwalys zero.

(ii) If each of the n given observation be doubled, then their mean is doubled

(iii) Ifx is the mean of x1, x2, , xn . the mean of ax1, x2, .. , axn is a x where a is any number

different form zero(iv) Arithmetic mean is independent of origin i.e. it is x effected by any change in origin.

3 Geometric mean:

(i) For ungrouped data G.M. = (x1 x2 x3.. xn)1/n or

G.M. = antilog

(ii) For grouped data G.M. = (xf1

1 xf22.. x

fnn)

1/N, where N = i=1

nfi

4 Harmonic mean Harmonic mean is reciprocal of arithmetic mean of reciprocals.

(i) For ungrouped data H.M. =

(ii) For grouped data H.M. =

5 Relation between A.M., G.M and H.M.

A.M. G.M. H.M.

Equation holds only when all the observations in the series are same

6 Msdian:

(a) Individual series (ungrouped data) : If data is raw, arrange in ascending or descending order and n be

the no. of observations . If n is odd, Median = value of ((n+1) / 2))th observation If n is even, median =

(1/2) [value of (n/2)th + value of ((n/2) + 1)th] observation.

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(b) Discrete series: First find cumulative frequencies of the variables arranged in ascending or

descending order and Mediain = {(n+1) / 2}th

observation, where n is cumulative frequency.

(c) Continuous distribution (grouped data)

(i) For series in ascending order

Median = e +

Where

e = Lower limit of the median class.

f = Frequency of the median class.

N = Sum of the all frequencies.

i = The width of the median class.

C = Cumulative frequency of the class preceding to median class.

(ii) For series in descending order Median = u -

Where u = upper limit of median class.

7 Mode:

(i) For individual series: In the case of individual series, the value which is repeated maximum number of

times is the mode of the series.

(ii) For discrete frequency distribution series: In the case of discrete frequency distribution, mode is the

value of the variate corresponding to the maximum frequency.

(iii) For continuous frequency distribution : first find the model class i.e. the class which has maximum

frequency for continuous series

Where

e1 = Lower limit of the model class.

f1 = Frequency of the model class.

f0 = Frequency of the class preceding mode class.

f2 = Frequency of the class succeeding model class.

i= Size of the model class.

8 Relation between mean, mode & median:

(i) In symmetrical distribution : mean = mode = median

(ii) In Moderately symmetrical distribution : mode = 3 median 2 mean

Measure of Dispersion:

The degree to which numerical data tend to spread about an average value is called variation or

dispersion popular methods of measure of dispersion.

(a) Individual series (ungrouped data)

1 Mean deviationMean deviation = (|x-S| / n)

Where n = number of terms, S = deviation of variety form mean mode , median

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(b) Continuous series (grouped data)

Note:

Mean deviation is the least when measured from the median.

2 Standerd Deviation :

S.D. () is the squere root of the arithmetic mean of the squares of the deviations of the terms from

their A.M.

(a) for individual series (ungrouped data)

where x = Arithmetic mean of the series N = Total frequency

(b) For continuous series (grouped data)

(i) Direct method =

Where

x = Arithmetic mean of series

X1 = mid value of the class

f1 = Frequency of the corresponding x1

N = f = Total frequency

(ii) Short cut method

Where

d = x - A = Derivation from assumed mean A

f = Frequency of item (term)

N = f = Total frequency.

Variance Square of standard direction

i.e. variance = (S.D.)2 = ()2

Coefficient of variance = Coefficient of S.D. x 100 = ( / x) x 100